Development of an Item Bank for Measuring Students’ Conceptual Understanding of Real Functions

Anela Hrnjičić 1 2 * , Adis Alihodžić 1, Fikret Čunjalo 1, Dina Kamber Hamzić 1
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1 Department of Mathematics, Faculty of Science, University of Sarajevo, Sarajevo, BOSNIA & HERZEGOVINA
2 Department of Mathematics and Informatics, Faculty of Philosophy, University of Zenica, Zenica, BOSNIA & HERZEGOVINA
* Corresponding Author
EUR J SCI MATH ED, Volume 10, Issue 4, pp. 455-470.
Published: 12 July 2022
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It is known that students have many misconceptions about concepts related to function. By discovering misconceptions using an appropriate measurement instrument, we can determine what changes we need to make in the real functions curriculum to improve learning outcomes. Therefore, we designed an item bank for measuring conceptual understandings of real functions with items that require ability to move from one representation of the same concept to another. By surveying university professors, we conducted an expert judgment and content validity of the test. Altogether 36 multiple-choice items based on concepts related to real function with a single correct answer and three distractors have been field-tested by means of a paper and pencil survey, which included 80 freshman students from the Faculty of Science at the University of Sarajevo. By surveying students, we checked technical characteristics of items and their cognitive validity. Results from surveying university professors and students show that the test meets the requirements of content and cognitive validity. Results from the item analysis (item difficulty index, item discrimination index, and point-biserial coefficient) and test analysis (test reliability and Ferguson’s delta) show that 32 out of 36 items have good psychometric characteristics, and they are reliable for measuring students’ understanding and skills in introductory mathematics courses at universities. We noticed that students have a poor understanding of certain concepts, regardless of the representation, and that there is no coordination between representations of the same concept.


Hrnjičić, A., Alihodžić, A., Čunjalo, F., & Kamber Hamzić, D. (2022). Development of an Item Bank for Measuring Students’ Conceptual Understanding of Real Functions. European Journal of Science and Mathematics Education, 10(4), 455-470.


  • APOSO. (2015). Zajednička jezgra nastavnih planova i programa za matematičko područje definisana na ishodima učenja [The common core mathematics curriculum based on learning outcomes]. Agency for Pre-Primary, Primary and Secondary Education, Bosnia and Herzegovina, Council of Ministers.
  • Bardelle, C., & Ferrari, P. L. (2011). Definitions and examples in elementary calculus: The case of monotonicity of functions. ZDM, 43(2), 233-246.
  • Bardini, C., Pierce, R., Vincent, J., & King, D. (2014). Undergraduate mathematics students’ understanding of the concept of function. Indonesian Mathematical Society Journal on Mathematics Education, 5(2), 85-107.
  • Bezuidenhout, J. (2001). Limits and continuity: Some conceptions of first-year students. International Journal of Mathematical Education in Science and Technology, 32(4), 487-500.
  • Bisson, M. J., Gilmore, C., Inglis, M., & Jones, I. (2016). Measuring conceptual understanding using comparative judgement. International Journal of Research in Undergraduate Mathematics Education, 2, 141-164.
  • Bjorner, J. B., Chang, C. H., Thissen, D., & Reeve, B. B. (2007). Developing tailored instruments: Item banking and computerized adaptive assessment. Quality of Life Research, 16(1), 95-108.
  • Carlson, M., Oehrtman, M., & Engelke, N. (2010). The precalculus concept assessment (PCA) instrument: A tool for assessing reasoning patterns, understandings and knowledge of precalculus level students. Cognition and Instruction, 28(2), 113-145.
  • Code, W., Piccolo, C., Kohler, D., & MacLean, M. (2014). Teaching methods comparison in a large calculus class. ZDM, 46, 589-601.
  • Crooks, N. M., & Alibali, M. W. (2014). Defining and measuring conceptual knowledge in mathematics. Developmental Review, 34(4), 344-377.
  • Day, J., & Bonn, D. (2011). Development of the concise data processing assessment. Physical Review Special Topics-Physics Education Research, 7(1), 010114.
  • Ding, L., & Beichner, R. (2009). Approaches to data analysis of multiple-choice questions. Physical Review Special Topics-Physics Education Research, 5(2), 020103.
  • Doran, R. L. (1980). Basic measurement and evaluation of science instruction. National Science Teachers Association, 1742 Connecticut Ave., NW, Washington, DC 20009 (Stock No. 471-14764; no price quoted).
  • Dreyfus, T., & Eisenberg, T. (1982). Intuitive functional concepts: A baseline study on intuitions. Journal for Research in Mathematics Education, 13(5), 360-380.
  • Ebel, R. L., & Frisbie, D. A. (1991). Essentials of educational measurement. University of Iowa.
  • Elia, I., & Spyrou, P. (2006). How students conceive function: A triarchic conceptual-semiotic model of the understanding of a complex concept. The Mathematics Enthusiast, 3(2), 256-272.
  • Even, R. (1990). Subject matter knowledge for teaching and the case of functions. Educational Studies in Mathematics, 21(6), 521-544.
  • Even, R. (1998). Factors involved in linking representations of functions. The Journal of Mathematical Behavior, 17, 105-121.
  • Gagatsis, A., & Shiakalli, M. (2004). Ability to translate from one representation of the concept of function to another and mathematical problem solving. Educational Psychology, 24(5), 645-657.
  • George, D., & Mallery, P. (2003). SPSS for Windows step by step: A simple guide and reference. Allyn& Bacon.
  • Habre, S., & Abboud, M. (2006). Students’ conceptual understanding of a function and its derivative in an experimental calculus course. The Journal of Mathematical Behavior, 25(1), 57-72.
  • Hiebert, J. (Ed.). (2013). Conceptual and procedural knowledge: The case of mathematics. Routledge.
  • Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1-27). Erlbaum.
  • Hitt, F. (1998). Difficulties in the articulation of different representations linked to the concept of function. The Journal of Mathematical Behavior, 17, 123-134.
  • Hornsby, E. J., & Cole, M. (1986). Rational functions: Ignored too long in the high school curriculum. Mathematics Teacher, 79(9), 691-697.
  • Husremović, Dž. (2016). Osnove psihometrije za studente psihologije [The basics of psychometrics for psychology students]. Faculty of Philosophy, University of Sarajevo.
  • Kalchman, M., & Koedinger, K. R. (2005). Teaching and learning functions. How Students Learn: History, Mathematics, and Science in the Classroom, 351-393.
  • Kline, P. (1986). A handbook of test construction: Introduction to psychometric design. Routledge.
  • Liu, X. (2010). Using and developing measurement instruments in science education: A Rasch modeling approach. IAP.
  • Lloyd, G., Beckmann, S., Zbiek, R. M., & Cooney, T. (2010). Developing essential understanding of functions for teaching mathematics in grades 9-12. National Council of Teachers of Mathematics. 1906 Association Drive, Reston, VA 20191-1502.
  • Mešić, V., Neumann, K., Aviani, I., Hasović, E., Boone, W. J., Erceg, N., Grubelink, V., Sušac, A., Glamočić, S. Dž, Karuza., M., Vidak, A., Alihodžić, A., & Repnik, R. (2019). Measuring students’ conceptual understanding of wave optics: A Rasch modelling approach. Physical Review Physics Education Research, 15(1), 010115.
  • National Research Council, Division of Behavioral and Social Sciences and Education, Center for Education, Mathematics Learning Study Committee. (2001). Adding it up: Helping children learn mathematics (J. Kilpatrick, J. Swafford, & B. Findell, Eds.). National Academy Press.
  • NCTM. (2000). Principles and standards for school mathematics. National Council of Teachers of Mathematics.
  • Nitsch, R., Fredebohm, A., Bruder, R., Kelava, A., Naccarella, D., Leuders, T., & Wirtz, M. (2015). Students’ competencies in working with functions in secondary mathematics education–Empirical examination of a competence structure model. International Journal of Science and Mathematics Education, 13(3), 657-682.
  • Orton, A. (1983). Students’ understanding of differentiation. Educational Studies in Mathematics, 14, 235-250.
  • Rittle-Johnson, B., & Schneider, M. (2015). Developing conceptual and procedural knowledge of mathematics. In R. C. Cohen, & A. Dowker (Eds.), Oxford handbook of numerical cognition (pp. 1102-1118). Oxford University Press.
  • Sánchez, V., & Llinares, S. (2003). Four student teachers’ pedagogical reasoning on functions. Journal of Mathematics Teacher Education, 6(1), 5-25.
  • Schneider, M., & Stern, E. (2010). The developmental relations between conceptual and procedural knowledge: A multimethod approach. Developmental Psychology, 46(1), 178-192.
  • Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1-36.
  • Sierpinska, A. (1992). On understanding the notion of function. In E. Dubinsky, & G. Harel (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 23-58). Mathematical Association of America.
  • Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77(1), 20-26.
  • Szydlik, J. (2000). Mathematical beliefs and conceptual understanding of the limit of a function. Journal for Research in Mathematics Education, 31(3), 258-276.
  • Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151-169.
  • Tarrant, M., Ware, J., & Mohammed, A. M. (2009). An assessment of functioning and non-functioning distractors in multiple-choice questions: A descriptive analysis. BMC Medical Education, 9(1), 1-8.
  • Urbina, S. (2004). Essentials of psychological testing. John C Wiley & Sons.
  • Zazkis, D. (2014). Proof-scripts as a lens for exploring students’ understanding of odd/even functions. The Journal of Mathematical Behavior, 35, 31-43.